Now that we’ve got the existence and uniqueness of our solutions down, we have one more of our promised results: the smooth dependence of solutions on initial conditions. That is, if we use our existence and uniqueness theorems to construct a unique “flow” function satisfying
by setting — where is the unique solution with initial condition — then is continuously differentiable.
Now, we already know that is continuously differentiable in the time direction by definition. What we need to show is that the directional derivatives involving directions in exist and are continuous. To that end, let be a base point and be a small enough displacement that as well. Similarly, let be a fixed point in time and let be a small change in time
But now our result from last time tells us that these solutions can diverge no faster than exponentially. Thus we conclude that
and so as this term must go to zero as well. Meanwhile, the second term also goes to zero by the differentiability of . We can now see that the directional derivative at in the direction of exists.
But are these directional derivatives continuous. This turns out to be a lot more messy, but essentially doable by similar methods and a generalization of Gronwall’s inequality. For the sake of getting back to differential equations I’m going to just assert that not only do all directional derivatives exist, but they’re continuous, and thus the flow is .